Hi all,
I emailed my query to tech support at Stata corp and below is the response;
Typically for a fixed effects negative binomial model, you would want to use
the -xtnbreg, fe- command. -xtnbreg, fe- is fitting a conditional fixed
effects model. When you include panel dummies in -nbreg- command, you are
fitting an unconditional fixed effects model. For nonlinear models such as
the negative binomial model, the unconditional fixed effects estimator
produces inconsistent estimates. This is caused by the incidental parameters
problem. See the following references for theoretical aspects on the
incidental parameters problem:
Greene, William H. "Econometric Analysis". Prentice Hall.
Seventh Edition, page 413.
Baltagi, Badi "Econometric Analysis of Panel Data".
4th. Edition. John Wiley and Sons LTD.
Section 11.1 (pages 237-8).

Here is the abstract for the Allison & Waterman paper I mentioned before:

"This paper demonstrates that the conditional negative binomial model
for panel data, proposed by Hausman, Hall, and Griliches (1984), is
not a true fixed-effects method. This method which has been
implemented in both Stata and LIMDEP-does not in fact control for all
stable covariates. Three alternative methods are explored. A negative
multinomial model yields the same estimator as the conditional
Poisson estimator and hence does not provide any additional leverage
for dealing with overdispersion. On the other hand, a simulation
study yields good results from applying an unconditional negative
binomial regression estimator with dummy variables to represent the
fixed effects.
There is no evidence for any incidental parameters bias in the
coefficients, and downward bias in the standard error estimates can
be easily and effectively corrected using the deviance statistic.
Finally, an approximate conditional method is found to perform at
about the same level as the unconditional estimator."

And, from the conclusion:

"The negative binomial model of Hausman, Hall, and Griliches (1984)
and its associated conditional likelihood estimator does not
accomplish what is usually desired in a fixed-effects method, the
control of all stable covariates. That is because the model is based
on a regression decomposition of the overdispersion parameter rather
than the usual regression decomposition of the mean. Symptomatic of
the problem is that programs that implement the conditional estimator
have no difficulty estimating an intercept or coefficients for
time-invariant covariates."

The empirical examples Allison provides (p. 64 of his book), for
which he says "None of this makes sense for a true fixed effects
estimator" seem pretty compelling to me, but I remain open to
persuasion or correction.